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The probability density function contains not only the first-order and the second-order statistics, but also the higher-order statistics and more complex feature information. Information fusion on probability density functions for multi-sensor is a challenging problem to be solved in the field of signal processing, especially in the application prospect of multi-sensor multi-scale information fusion for automatic driving and unmanned system, this problem has gradually attracted much attention. How to design fusion criteria and how to form a unified fusion framework are the to-pics that scientists and engineers are committed to solving. Aiming at the fusion of multiple probability density functions from multi-sensor of random variables (vectors), this paper investigates the existing relevant fusion theories and methods, and provides some design rules, criteria, principles and theorems, such as axiomatic method, optimization method and super Bayesian method. The study is expected to provide some directional guide for the effective solution of the fusion problem.

This article proposes two robust Kalman filters to solve the issue of inaccurate modeling in multiplicative noise systems due to epistemic limitations. First, we construct all conceivable state/measurement transition probability densities as an ambiguity set. This ambiguity set chooses the Wasserstein distance or the moment‐based metric as the distance metric. Besides, this set is an inequality set with a chosen tolerance, which can be seen as a non‐negative radius ball. Then, by combining the robust solution of the least favorable model in that ball with the alternating direction method of multipliers or an efficient direct solution method, we propose two robust Kalman filters based on the minimum mean square error criterion. A classical example is provided to verify the effectiveness of the proposed robust filters in comparison to existing state‐of‐the‐art filters. [ABSTRACT FROM AUTHOR]

This article focuses on the adaptive Kalman filtering problem for linear systems with unknown covariances of both dynamic multiplicative noise (multiplicative measurement noise) and additive noises (additive process and measurement noises). A recursive-noise adaptive Kalman filter is proposed to estimate both states and covariances of noises by using the variational Bayesian (VB) inference and an indirect method. First, we characterize inverse Wishart priors for both measurement noise covariance and process noise covariance and employ the Student’s t-distribution to represent the likelihood function, which is non-Gaussian and affected by mixing multiplicative noise and additive measurement noise. Then, an adaptive Kalman filtering for recursive both noise covariance matrices and dynamic state is proposed following VB inference. Performance analysis for VB procedures and the proposed filter is provided to ensure the convergence and stability. A target tracking example is provided to validate the effectiveness of the proposed filtering algorithm. [ABSTRACT FROM AUTHOR]

Summary: This article considers a distributed Kalman filtering problem for linear system contaminated by complex multi‐channel random uncertain parameter in which a number of nodes cooperative without central coordination to estimate a common state based on local measurements and data received from neighbors. We propose an approach to eliminate this error propagation. The proposed local filters are guaranteed to be stable under some mild conditions on certain global structural data, and their fusion yields the centralized Kalman estimate. Then, we extend this method to smoothing and deconvolution algorithm. Finally, simulation experiments demonstrate the validity of the proposed approach. [ABSTRACT FROM AUTHOR]

This brief proposes an optimal filtering algorithm for linear singular systems with stochastic multiplicative noise by using the second restricted equivalent transformation (SRET) to reduce computation load, avoid curse of dimensionality and improve precision. The presented filter can be guarantee optimum in the sense of linear minimum mean-square error (LMMSE) by employing the projection theorem. A performance analysis and simulation results verify the proposed algorithm. [ABSTRACT FROM AUTHOR]

Summary: This article considers robust fusion algorithms for linear system with multichannel stochastic multiplicative disturbance by using projection theorem and projection formula. Based on the centralized and distributed (feedback and no feedback) fusion strategies, we present optimal filtering fusion algorithms, optimal fixed‐interval smoothing fusion algorithms, and fixed‐interval deconvolution fusion algorithms. The methods of projection theorem and projection formula are used to guarantee the optimality of these derived algorithms. The presented algorithms are much less computation burden and higher accuracy than existing state‐of‐the‐art methods. Numerical simulation results including a WSN network model verify the proposed algorithms. [ABSTRACT FROM AUTHOR]

Summary: This article studies singular systems with uncertain parameters by using a direct method which is called the first restricted equivalent transformation (FRET). The stochastic uncertain parameters we studied in the article, which are more general and challenging, are different from the two existing types: time‐invariant parameters and time‐varying parameters. We propose an optimal filtering algorithm to filter out artifacts, such as noise, from the uncertain parameter by using the projection theorem and the FRET. The proposed algorithm guarantees optimum in the sense of linear minimum mean‐square error. Simulation results verify the proposed algorithm. [ABSTRACT FROM AUTHOR]

This brief considers estimation algorithms for linear singular systems with random uncertain parameter matrices. We propose an optimal filtering algorithm and an optimal fixed-interval smoothing algorithm to filter out artifacts, such as noise, from the parameter matrix by using the projection theorem and an equivalent transformation. A performance analysis and simulation results verify the proposed algorithms. [ABSTRACT FROM AUTHOR]

The paper considers target tracking system with multiplicative noise. Gaussian white, colored and non-Gaussian multiplicative noise are taken into account. We present optimal filtering algorithms to settle tracking system contaminated by Gaussian white and colored multiplicative noise and a suboptimal filtering algorithm to tackle the non-Gaussian case. The proposed optimal filtering algorithms are derived by using projection theorem and the suboptimal filtering algorithm is deduced by applying variational Bayesian inference. Simulation results verify the effectiveness of these presented algorithms. [ABSTRACT FROM AUTHOR]

Summary: This paper considers estimation algorithms for linear and nonlinear systems contaminated by non‐Gaussian multiplicative and additive noises. Based on the variational idea, in order to derive optimal estimation algorithms, we combine the multiplicative noise with states as the joint parameters to estimate. The application of variational Bayesian inference to joint estimation of the state and the multiplicative noise is established. By treating the states as unknown quantities as well as the multiplicative noise, there are now correlations between the states and multiplicative noise in the posterior distribution. There are two main goals in Bayesian learning. The first is approximating the marginal likelihood (PDF of multiplicative noise) to perform model comparison. The second is approximating the posterior distribution over the states (also called a system model), which can then be used for prediction. The two goals constitute the iterative algorithm. The rules for determining the loop is the Kullback‐Leibler divergence between the true distribution of state and a chosen fixed tractable distribution, which is used to approximate the true one. The iterative algorithm is deduced, which is initialized based on the idea of sampling. Meanwhile, the convergence analysis of the proposed iterative algorithm is presented. The numerical simulation results in a comparison between the proposed method and these existing classic algorithms in the context of nonlinear hidden Markov models, state‐space models, and target‐tracking models with non‐Gaussian multiplicative noise demonstrate the superiorities, not only in speed, precision, and computation load but also in the ability to process non‐Gaussian complex noise. [ABSTRACT FROM AUTHOR]

Summary: Accurate system description is important to obtain accurate state estimation for many application areas. Much a priori information and complex noise pollution exist in real systems and impact the accuracy of system descriptions. Thus, this paper focuses on the state estimation problem with prior information for systems with complex noise pollution. Considering the additional prior information about the state with constraint increase, the estimate quality is compared to classical state estimation methods, which cannot utilize the prior information. Therefore, linear systems with complex noise include additive noise and multiplicative noise, which are taken into account, and an optimal algorithm is deduced. For nonlinear systems and non‐Gaussian complex noise, an algorithm is proposed based on variational Bayesian inference to address this general problem. Furthermore, an algorithm comparison, a performance analysis, and a convergence analysis are presented. The estimation performance is illustrated in numerical target tracking examples. [ABSTRACT FROM AUTHOR]

Summary: This paper studies an adaptive algorithm for the estimation problem of nonlinear systems with unknown or missing measurement noise and uncertain parameters using variational Bayesian (VB) inference. We combine VB inference with the Monte Carlo sampling technique to settle this problem. There are many cases of missing information, and because of the difficulty in obtaining the analytical results, the existing control methods for uncertain systems lack generality. We present a set of nonlinear recursive adaptive filtering algorithms that address the unknown parameters and probability density function. The proposed algorithms yield a separable variational approximation of the joint posterior distribution of noise parameters with uncertain parameters and states on each step separately. Estimation convergence and robustness against disturbances are guaranteed. A convergence result for VB inference is presented. Extensive simulation examples are provided to demonstrate the efficacy of the proposed algorithms. [ABSTRACT FROM AUTHOR]